Question: $\dfrac{ -6t - u }{ -3 } = \dfrac{ -10t - v }{ 8 }$ Solve for $t$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -6t - u }{ -{3} } = \dfrac{ -10t - v }{ 8 }$ $-{3} \cdot \dfrac{ -6t - u }{ -{3} } = -{3} \cdot \dfrac{ -10t - v }{ 8 }$ $-6t - u = -{3} \cdot \dfrac { -10t - v }{ 8 }$ Multiply both sides by the right denominator. $-6t - u = -3 \cdot \dfrac{ -10t - v }{ {8} }$ ${8} \cdot \left( -6t - u \right) = {8} \cdot -3 \cdot \dfrac{ -10t - v }{ {8} }$ ${8} \cdot \left( -6t - u \right) = -3 \cdot \left( -10t - v \right)$ Distribute both sides ${8} \cdot \left( -6t - u \right) = -{3} \cdot \left( -10t - v \right)$ $-{48}t - {8}u = {30}t + {3}v$ Combine $t$ terms on the left. $-{48t} - 8u = {30t} + 3v$ $-{78t} - 8u = 3v$ Move the $u$ term to the right. $-78t - {8u} = 3v$ $-78t = 3v + {8u}$ Isolate $t$ by dividing both sides by its coefficient. $-{78}t = 3v + 8u$ $t = \dfrac{ 3v + 8u }{ -{78} }$ Swap signs so the denominator isn't negative. $t = \dfrac{ -{3}v - {8}u }{ {78} }$